A New Lower Bound for the Diagonal Poset Ramsey Numbers

TL;DR

This paper establishes a new linear lower bound for the diagonal poset Ramsey number $R(Q_n,Q_n)$, improving previous bounds and suggesting it could be close to three times $n$ with advanced layered-coloring techniques.

Contribution

The paper introduces a stronger, more constructive method to improve the lower bound of $R(Q_n,Q_n)$ from 2.02n to at least 2.7n, with potential to reach nearly 3n.

Findings

Proved $R(Q_n,Q_n) \\geq 2.7n + k$ for some constant k.

Developed layered-coloring modifications that surpass probabilistic approaches.

Indicated the possibility of approaching a lower bound of approximately 3n.

Abstract

Given two finite posets $\mathcal P$ and $\mathcal Q$, their Ramsey number, denoted by $R(\mathcal P,\mathcal Q)$, is defined to be the smallest integer $N$ such that any blue/red colouring of the vertices of the hypercube $Q_N$ has either a blue induced copy of $\mathcal P$, or a red induced copy of $\mathcal Q$. Axenovich and Walzer showed that, for fixed $\mathcal P$, $R(\mathcal P, Q_n)$ grows linearly with $n$. However, for the diagonal question, we do not even come close to knowing the order of growth of $R(Q_n,Q_n)$. The current upper bound is $R(Q_n,Q_n)\leq n^2-(1-o(1))n\log n$, due to Axenovich and Winter. What about lower bounds? It is trivial to see that $2n\leq R(Q_n,Q_n)$, but surprisingly, even an incremental improvement required significant work. Recently, an elegant probabilistic argument of Winter gave that, for large enough $n$, $R(Q_n,Q_n)\geq 2.02n$. In this paper we show that $R(Q_n,Q_n)\geq 2.7n+k$, where $k$ is a constant. Our current techniques might in principle show that in fact, for every $\epsilon>0$, for large enough $n$, $R(Q_n,Q_n)\geq (3-\epsilon)n$. Our methods exploit careful modifications of layered-colourings, for a large number of layers. These modifications are stronger than previous arguments as they are more constructive, rather than purely probabilistic.